Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $z \neq 0$. $n = \dfrac{z^2 + 11z + 30}{6z + 60} \div \dfrac{-3z^2 - 15z}{7z + 70} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $n = \dfrac{z^2 + 11z + 30}{6z + 60} \times \dfrac{7z + 70}{-3z^2 - 15z} $ First factor the quadratic. $n = \dfrac{(z + 5)(z + 6)}{6z + 60} \times \dfrac{7z + 70}{-3z^2 - 15z} $ Then factor out any other terms. $n = \dfrac{(z + 5)(z + 6)}{6(z + 10)} \times \dfrac{7(z + 10)}{-3z(z + 5)} $ Then multiply the two numerators and multiply the two denominators. $n = \dfrac{ (z + 5)(z + 6) \times 7(z + 10) } { 6(z + 10) \times -3z(z + 5) } $ $n = \dfrac{ 7(z + 5)(z + 6)(z + 10)}{ -18z(z + 10)(z + 5)} $ Notice that $(z + 10)$ and $(z + 5)$ appear in both the numerator and denominator so we can cancel them. $n = \dfrac{ 7\cancel{(z + 5)}(z + 6)(z + 10)}{ -18z(z + 10)\cancel{(z + 5)}} $ We are dividing by $z + 5$ , so $z + 5 \neq 0$ Therefore, $z \neq -5$ $n = \dfrac{ 7\cancel{(z + 5)}(z + 6)\cancel{(z + 10)}}{ -18z\cancel{(z + 10)}\cancel{(z + 5)}} $ We are dividing by $z + 10$ , so $z + 10 \neq 0$ Therefore, $z \neq -10$ $n = \dfrac{7(z + 6)}{-18z} $ $n = \dfrac{-7(z + 6)}{18z} ; \space z \neq -5 ; \space z \neq -10 $